We investigate the problem of assembling general shapes and patterns in a model in which particles move based on uniform external forces until they encounter an obstacle. In this model, corresponding particles may bond when adjacent with one another. Succinctly, this model considers a 2D grid of “open” and “blocked” spaces, along with a set of slidable polyominoes placed at open locations on the board. The board may be tilted in any of the 4 cardinal directions, causing all slidable polyominoes to move maximally in the specified direction until blocked. By successively applying a sequence of such tilts, along with allowing different polyominoes to stick when adjacent, tilt sequences provide a method to reconfigure an initial board configuration so as to assemble a collection of previous separate polyominoes into a larger shape.
While previous work within this model of assembly has focused on designing a specific board configuration for the assembly of a specific given shape, we propose the problem of designing universal configurations that are capable of constructing a large class of shapes and patterns. For these constructions, we present the notions of weak and strong universality which indicate the presence of “excess” polyominoes after the shape is constructed. In particular, for given integers h, w, we show that there exists a weakly universal configuration with O(hw) 1 × 1 slidable particles that can be reconfigured to build any h × w patterned rectangle. We then expand this result to show that there exists a weakly universal configuration that can build any h × w-bounded size connected shape. Following these results, which require an admittedly relaxed assembly definition, we go on to show the existence of a strongly universal configuration (no excess particles) which can assemble any shape within a previously studied “drop” class, while using quadratically less space than previous results.
Finally, we include a study of the complexity of deciding if a particle within a configuration may be relocated to another position, and deciding if a given configuration may be transformed into a second given configuration. We show both problems to be PSPACE-complete even when no particles stick to one another and movable particles are restricted to 1 × 1 tiles and a single 2 × 2 polyomino.
Balanza-Martinez, J., Luchsinger, A., Caballero, D., Reyes, R., Cantu, A. A., Schweller, R., Garcia, L. A., & Wylie, T. (2019). Full Tilt: Universal Constructors for General Shapes with Uniform External Forces. In Proceedings of the 2019 Annual ACM-SIAM Symposium on Discrete Algorithms (Vol. 1–0, pp. 2689–2708). Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611975482.167
In Proceedings of the 2019 Annual ACM-SIAM Symposium on Discrete Algorithms (Vol. 1–0, pp. 2689–2708)